# 2-shifted Poisson bracket on Lie algebra cohomology

Let $$\frak{g}$$ be a semisimple Lie algebra, and let $$({-},{-})$$ be an invariant inner product on $$\frak{g}$$. The Chevalley–Eilenberg complex $$C^*(\frak{g})$$ has a natural Poisson bracket of degree $$-2$$; if we think of the generators of the exterior algebra $$C^*(\frak{g})$$ as coordinates $$\xi$$ on a graded manifold, then this Poisson bracket is associated to the symplectic form $$\Omega=(d\xi,d\xi)$$.

The resulting (shifted) differential graded Lie algebra is formal, and the Poisson bracket induces the vanishing bracket on the cohomology. I have written down a proof, which uses explicit generators for the cohomology (essentially, the Chern–Simons classes). Is there a published proof of this result, perhaps less computational?

After posting this question, I learned that this construction is discussed in the article

Shifted Poisson and symplectic structures on derived N-stacks Jon Pridham (Examples 3.31)

Unless I am mistaken, the special case where $$\mathfrak{g}$$ is semisimple is not addressed in Pridham's article. I am grateful for all of the general bibliographic references, but I am interested in a very specific result, and none of the comments below address the question I asked.

• Something doesn’t smell right. What if g is abelian? Then the differential is zero, so it is automatically formal, but the bracket is not zero. Oct 13, 2021 at 21:24
• Those co-ordinates are in the wrong degree to give a derived affine scheme. That's a form of Lie algebroid, enhancing things in the opposite direction, so $H^*$-isomorphism is too weak to give an equivalence on the associated stacks. Oct 13, 2021 at 21:27
• Isn’t this the standard 2-shifted symplectic structure on BG,(associated to an invariant inner product) pulled back to its formal completion (B exp(g) )? Oct 13, 2021 at 22:58
• @DavidBen-Zvi : yes, as in Examples 3.31 of arxiv.org/abs/1504.01940 , for instance. Oct 13, 2021 at 23:14
• The paper of Pridham cited is a terrific reference for a very general construction including this one - but the explicit example in the question (for reductive groups) is already in the introduction of the original Pantev-Toën-Vaquié-Vezzosi paper on shifted symplectic structures (they say it for the group, but here we're just taking the formal completion of their example). Oct 14, 2021 at 15:41

I still hope that someone will answer this question. In the meantime, I will post a sketch of the proof that I found. It uses Chevalley's theorem, that the space of invariant polynomials $$I(\mathfrak{g})$$ of a reductive Lie algebra is spanned by the trace polynomials $$\text{Tr}_V(\rho(x)^k)$$, where $$\rho:\mathfrak{g}\to V$$ is a finite dimensional representation of $$\mathfrak{g}$$. This implies 2 things: 1) for any invariant polynomial, $$P(\frac12[\theta,\theta])$$ vanishes, where $$\theta\in C^1(\mathfrak{g},\mathfrak{g})$$ is the Maurer-Cartan form; and 2) the algebra $$C^*(\mathfrak{g})$$ is generated by derivatives $$(\theta,\nabla P(\frac12[\theta,\theta]))$$. The vanishing of the Poisson bracket on $$C^*(\mathfrak{g})^{\mathfrak{g}}$$ follows.